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Full text of "Nuclear Physics by Enrico Fermi: Chapter 3"

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Chapter II± 

A Q,^h„, A- Penetration of Pen tangular Earrier. 
of alpha" emS loT Sde^dS] ?£* ZZT ^J^T^^ ^™ 
don in ic 28 . This expStion^Ses^u I M aSdVS '° n " 
SS2 U SrS^f?J^^?_ t » .Pe»t»t» a po^LKallarSer . 

first the problem of the one dimensional 
rectangular harrier will he considered. 

Classically an Incident 
particle of energy E (see Pig. 
II 1. 1), coming from the left/" 
could not penetrate into re -Ion 
II, since U > E. It would be 
completely reflected. However, 


FIG- . Ill . 1 

quantum mechanically, the oarticle wh+iMn ■*« ■ 
ia nerriqiiv -r^fit^^L.i ^ P ■ ' whieh ls an incident wave, 

par ' lalI; reflected ana partially transmitted. 

It *f ^^^particle travelling to the 

Let f =dp Li 


urn %^Q 


is b.h'e reflected wave 


and ^ ^/€ i-g the transmitted wave. 

The probability of penetration or the 

w» rtt3 ,. transm itted intensity HPT 

transparency - iHHHiHt"intenHt'y * l^r 

Considering the time Independent Dart of/ 

re 1 

t &-, 

These are solutions of the time Independent wave equate 

d ■%*■ * 

In region II the wave equation becomes 

^jjj-i ft* t'H c // —° 
Lr:t B ~U-E 

Then g„KgV*3* ^ L g-^- 


phfsicallV " — tn& WaYe -^tion must be well behaved 

Thus $ and || are cort: 
x - b. 

nuous across the boundaries x = and 


56 Rectangular barrier Ch. Ill 

Thus 4- +/3= K +l since % (0) =$'{.*} 

m^ , c^ & )^ute 

MbM u - l 6 

Eliminating a between the first two of these equations gives 
Solving the last two of these equations for K and L gives 


L .x £•"'-(" f)e « 



* = £ 


■T - 1 

*=„£*?, A^jjF ~, 

transparency ^ ^ 





Notice that as h -*■ 0, this 
approaches the classical limit of 
zero transparency. Also the trans- 
parency is greater for particles 
of smaller mass. This explains why 
electrons have little trouble in 
going from one atom to another in 
diatomic molecules and metals. 
See PIG III. 2. 



PIG HI, 2 Diatomic molecule 
with one electron 

B. Barrier of Arbitrary Shape 

The order of magnitude of the 
transparency of an arbitrarily 
shaped barrier can be obtained by 
finding the average "height" and 
treating as a rectangular barrier. 


Ch. Ill 

III.l then gives 

Rectangular barrier 



'^Mfa^l ^ 


III. 2 


bump In 
100 ft. 
get past 
the c ar 

A very slowly moving car of 1 ton (kinetic energy con- 
almosL zero) encounters a sinusoidal T , M 
the road which Is 1 ft. high and 
long. Classically, the car can't 
However 112,2 shows that there 
is a finite probability (w) that 
can overcome the bump. 

TJ(x) = jjg x ig x 2.54 3in uU-aJ_ 



U(x) dx ^ 5.3 x 10 J 





-L.Z *t&~ 


In the followine figure, III .2 is used to calculate the 
transparency for Protons penetrating the coulomb barrier of nuclei 
ot A = 1, a, 90, anci 238. These are slotted as a function of 
troton energy divided by barrier energy. 





a*(«Rtia? £T 

















Trans oar encv to Protons of Nuclear Coulomb Barriers 


Alpha Decay 

Ch. Ill 

C, Serai-classical Application to Alpha Decay 
III. 2 will now be applied to calculate the probability of an 


alpha particle penetrating the potential 
barrier due to a nucleus. First, assump- 
tions must be made concerning this barrier. 
As an alpha particle Is brought from r=oo, 
the main interaction force is the electro- 
static repulsion which gives the potential 
U(r) =3Zgr . Let R be the distance where 
the alpha particle first "contacts" the 
nucleus. Then It encounters the strong 
short range attractive forces until it Is 
within the nucleus. Once the alpha part- pig III. 4 
icle Is in the nucleus it is considered as a free particle. The 
depth of this well is not very great for the alpha active nuclei, 
since the lowest energy state "must give a positive erergy eigen-' 
value. The height of the barrier is much greater, however. For 
uranium it is 29 Mev if equation 1.4, page 6, is used for R. 

probability / sec. of escape = rate of hitting barrier x 
transparency or T^r— 1 , 

f "R~ x fc- e fe * where /j, is the re- 

duced mass of alpha 
particle and nucleus 

7=r ^ tO since V lr/ v 10 and p-v/o - ' 1 

where & = ,EJt 

^jk*^f«TJ5) -sT) 

j ~'°" «gW*VjHg-4FF >j 


For a large br-arrier G has the following approximate form: 

= \Tv / where V is the final velocity of the °L particle 

Problem : Using formula I. 8, page 7, for the masses, A 

net ermine which of the following nuclei are unstable loo" 

to alpha decay and determine their lifetimes. 150 62 

For the first two nuclei listed, the mass of 200 80 

the daughter exceeds the mass of the parent, hence £50 97 

they are stable. Of the next three, Z=BO has the 300 113 
longest lifetime and will be calculated in detail 

11(200, 80) = lit 198, 7 8) + Ke 4 + $ M( 200, 80) =200. 04474 

■ , „ ,, M( 196, 78 ) =196.03826 

Q a 14(200,80) - p,l( 196,7 8) + He^J He*- 4.00390 

$ =- 0.00258 am = 3.84 x 10 -6 ergs 
zZe 2 

- 3,84 x 10-°ergs 

b = 9.52 x 10-l^cm 


= 1.5 x 10~ 15 {196) 3 = 8.7 x 10-l 3 cm bv 1.4 

Alpha decay 
III. 3 gives for G: 

= 127 x 1,198 

= 152 
£ -v fa**^-**-* 

t - 3 x 1035 vra . since 3>15 ^ iq7 sec = ^ ^ 

?hus this nucleus would appear stable, 


T>. Virtual Level Theory of Alpha Decay 

Now a complete quantum mechanical rresen^Hn. , m v. 
given, includlrs- the , n i„f=A^ £ ., presentation will be 
nucleus. This lladsto h vtf,° 7^ ^ pha P"^°l« inside the 

Problem ; At 2 = there is an 

infinite barrier. The onlv 
finite solution In this region 
is zero 

since -Sp? 




Let i>*= 'Jr 



for x £ 0. 

In regions I and III d^ + ^ 

Since ^ must be zero in the Jnfi'nWa «** 4-- , 

ffiust be continuous. ml h*S infi - lte potential region anfl / 

+ t>* 

#- e 

v e ha ye 

where the arbitrary constant Is 
chosen as one. 



< P&C-4-) 


60 Virtual level theory 

The continuity conditions at the boundaries give 

since *&.&>=*.# AJ 

Let = 2q(b-a) 

*V->c 4 = C 

: + e 


_ & 

#>*> -g'^ 

4^) - fei 

Ch, III 

III. 4 

j&tfa pfiz _ & yj 

than S arV^Hr lntenslt y of ^ *•■ usually much greater 
than thaj of ^ . however we are interested In the case where 
r^ is aa snail as possible, 

II 1. 4 gives C + = iiLemps* +. ^£-A^ra^v . 

5 T ~k- * * 


The values of E which satisfy 
the above equation give the small- 
est possible intensity outside. 
These values of E are known as the 
virtual levels. It is seen the out- 
side intensity decreases very sharply 
In ilie energy region of the virtual 

renera! 1 sSuaMo^S fo }} ^ W* the first curve shows the 
§!£? - flvJ n Wh ^ n C+ f °' The second curve is for the vir- 

fo^si T f?l Th6 fi2ial curve shows ^e behavior of ^ 

lor an energy sllgntly greater than the virtual level energy 

Ch. Ill 

Virtual level theory 

This same virtual level 
approach will now be used 
with the W.K.B. method In 
solving the ppdblsBi of the 
potential barrier- experienced 
by tlie r.l.Tiiia nsrT,i cle . 




FIG 7 gives the potential barrier experienced by the alpha 
aruicle. Since this Is a three dimensional problem, spherical 
oordmates vfill be used which give II 1. 5 for the radial part 


of the wave function. 

The complete wave equation is 

Let $m t g^) = fjpq Y tr jw) 

than ^[a^)] +>[*$(£- i^f) - l^jMw =0 

for regions II and III 

III. 5 

First the case where the angular -omentum of the alpha narti cle 
with respect to tne nucleus Is zero (s-state, £= 0} shall bs 

0^Y^4 e >=* 

where «.=A,^0 
for region I 
for real on II 

for region III 

III. 6 

Since these are all positive quantit ies , T , g, & , and $ are all 
u x = sirjr since £f#* ^/^ must be finite 

62 Virtual level theory Ch . ni 

The W.ff.B. method gives for the general solution in II 

a^o!; K r B : Tir^Jll^tT connectlng "«W***« *«"«. 

fce*?^ J'^^^^^Fj 




t^i^^aM G Is defin 

ed by III. 3 


This gives for u^ 

At large r %%*?** * J 

*t-tfee- t +4(?(r} 

III. 7 

SlatedTTh^ V*" Chapter > *■«« 67 > C_ and G + will be cal- 
culated In the region near a virtual level with the result 

^- ^W where fC^^J^(iXR)-£} 

JS***! J" 3 X* ener S^ reparation from the nearby virtual level 
and n Is tne separation of two adjacent virtual level's. 

Ill, 7 becomes 


III. 9 


a*« -* 

Thus except for a small energy range about the virtual level 

£ , S£ t £\£ t 3&iI s^ 1 " 8 " th *» ttal ^° ^ £ ole . 

tw* -**$*+- 





for i i ™ S 2<!r ° - I ' robttbl lity or zero amplitude 



:h. Ill 

Virtual level theory 


E n . ^ U) ln IIIj a11 corresponding to the same energy 

vlrLS'ene'rpv ^he°dffi^ 5*** f Mr W 8 W*™ «*»* the 

virtual energy. The distribution will turn out to he such that 
S^ will decrease exponentially with + ime HXa r>l+Z * f 

In summary the procedure will be- 

C1) So V?r?jS°?eSl? 1S ^ fuActi ° nS ln 6ne ^ «*i°* ab0 ^ 
fE) Expansion of ^(r,0) in terms of these eigen function 
(3) Determination of ra^e of decrease of ^ ? ^ unctlona - 

(1) Actually there is a continuous 
spectrum of eigen values and func- 
tions. The problem will first be 
treated as one of discrete levels 
by limiting region III to r< L. 
Later L will be made to approach 

infinity, which approaches -che continuous spectrum situation 
The region is lifted to r < L by placing an infinite barrLr'at 
- - L - L - lS taken large enough so that: 



/*£*& »/^^ 

IZXVTo'A,'* ; ff60il ' el >' " «>»• •«« (of n h.U-,av. 

■4 K ~ %J* ~ z 

£"^SiS^^ 7 ?.sa2r d - nornaiized " hioh 8i *«° » 


/<H.A.~ i 

III. 9 gives for the amulitude inside- 
amplitude inside 


The shape of the function u„ up to r - d fqe P w fil i„ D „ 

Inl G al 8 y the% Same /i°? ^ ^^ 1-elYnear ^%™u2 1 v T. 
±n hxg ti the smooth line represents 

un which has an E n slightly below 

the lowest virtual level, E , and 

the dotted line is for an enero-v 

slightly above S . 

Let f(r) (of amplitude one) 
represent all these u n 's for the 
region r< d. 

Then u n (r) 

if to 

for r< d 

III. 30 


Virtual level theory 

Ch. Ill 

If ^ is the wavelength of u n outside: rft-^p'sss-l. 

i. «. i& ~ A- 

3* IF 


-£. -i.2^^4.^ 

M*^i m "■** >t !> 

III. 11 

III. 12 

(2) Expansion of ¥ (r,0) in terms of u n . 

Up to nov^ f(r) has teen defined only for r<d. For r > d it 
will he defined = 0. 

Then^frjO) = ffr) so that the alpha particle has zero probabil- 
ity to he outside the nucleus at t = 0, 

since f(r) = for r > d 

c = 

_ J¥ 

by ill. 10 


This summation over n approaches 



using u n fr) for 
r < d from III. 10 

as L-»°° 

With the help of II I. 12 the variable of integration can be trans- 
formed: -t 

Since the integrand has large values only over a small range of 
n and n Is very large, n may be taken outside the integral. 

^JLfey &*&$££. f^" j. 


using III. 11 to eliminate n 

Ch * TI1 Virtual level theory 

where ^-f /g" 5 


= tfer Q 

-, PO _ i.APL 


- W3 — t-LK^- 


f5 ' cJ r,t ' decreaSQ s to ^ of /^ a (r,o) at 



which Is —-5- g S 

after substituting the values of A and B 
from page 62 

III. 13 

T ~—fk- ~€ 1Vhere E in = E - U Is the kinetic energy in. 

s; the nucleus. ■ * 

since V^ 




Ipproachf S Same rSSUlt " II1,3 Which U3ed a «ffll-dlw a ic«i 
Comparison with Experiment 

order? 1 of Ja^i?*^™*^* ° h9 2 kS Wlth e *P s ™^ as far as 
lifetime mlv^fS r c °£ cerned ' T he theoretical value of the 
103 Tir?hf HZ ^ th \ ex P e Cental by such a factor as 
10 . Thus the formula Is not very useful for computing life- 
times however, slncetis very sensitive to the radiuf R which 
appears In the exponent, a rough knowledge of \ will ^ive a Such 
.ore exact knowledge of R. In fact the j^* t ^ g™.^ 
~* D x 1U i0 A^ Is determined by averaging 'the radii nf ninho 

E B S B hl s C ect- e C R a « CITl u ted USi ^ thiS lh6 ^ Sli if aS? 
i^ It* l sec 1 tlon 6S.. No correction for the radius of the 
alpha particle is made In this determination. Be the gives two 

| Scent ^neXlt" °! d " ^ thS " neW " Whlch flif S S SoS 4.0 
Kl,iS.,? l^ arS m ° re in favor toda ^ Lethe's 

nev values are much larger because he considers that for most 
o. .he time the alpha particle does not exist separately L the 

Alpha decay, experimental 

^ .■-'.'■■ J. i f ' i.i :" J O '-.' .—.■:■ :■- .- ■■-. J 1 1. ~1 

Ch. Ill 

point cf ohe Gamow theory. This cpcibaMUH „? main weak 

the alnha r^i-vin -it i-t i prooaoility of pre-existance of 

d " tor lb „ ( -R" I to 10 lb J which gives a smaller harrier n^ 

larger radius for the same lifetime? " " 

Another consideration which has 
oeen neglected thus far is the possl 
cility of the alpha particle being 
emitted with angular momentum. The 
same mathematical treatment may be 
used if j£ 1JW lg added to t £ 

potential s ~ as can he seen by 
III. 5. This will increase the 
harrier and naif -life. See FIG 9. 
The effect Is not very great since 
an a as large as 5 changes r only 
by a factor of 10. 


The Geiger-JIuttall Law 

lonJ^J^i ael £ er ^fl Nut tall discovered the exoerirre-ntal re- 
lationship between members of the same radio-aotSe^Sr 

wnere R is the range in air III. 14 
For the 3 radio-active families this const. Is 

10-42.3 for , araniuj:]1 faEllly 
10-44.?. for thorium 

Since in the energvregion for alphafulo" iS ife^™ v3 „ 

gives a relationship between ^ and the velocity Tin', ™Ti o?V 
checks with that given by the" Gamow Seoly (II? 3)^ve/£e 
limited energy range which is observed. 

E. Alpha Kay Spectra 

„„.„ f Many al f*" a eailtters 3h0 ' a fine structure in the enerev 
spectrum of the emitted ainhaq. ™ j -m- k*-i ^ . Z e - ier &y 

tSsit^of'a^lvS'r- ^ °™ the0r ^ Sives thf ratio of^n- 
stZl L " 8 £g Gnergy SrOUP t0 that <* *he maximum energy 

<TE is the energy difference and E Q ia :.i., e 

maximum energy. 


general +p P nri nf 1 ^,™TZ , " ilLL ^O J-* ^ne maximum ene 

l^^ V ^jL^°fioTtSi e rJ^r rt3 th V iS Nation. The in 
ing dlffereni^ab^^ 
particle within the nucleus. existance of uhe alpha 

Gii. Ill 




1 . 

site* L - 
















Alpha Spectra 67 

pare nt daughter 

PIG- III. 11 

FIG- 11 la an energy level diagram Illustrating the phenom- 
enon ol" discrete energy groups, <*, Is the maximum energy alpha. 
The excited nuclear states of the daughter decay rapidly by 
gamma emission to the ground state,, E , where hv = E-t - E , etc. 
The energies of some of these gamma rays have he en observed and 
do check. 


Long Range Alpha Emission 

Two nuclei, RaC and ThC ' , give an alpha spectrum which 
contains low intensity groups of alphas up to 2 Mev ahove the 
main group. The intensities of 
these long range groups are from 
10-4 to 10 -6 of the total Inten- 
sity, This is because both RaC 
and ThC are products of a beta 
disintegration which leaves them 
in excited states. These excited 
states have a choice of decaying by 
a gamma ray to the ground parent state, 
or else by alpha to the daughter 
nucleus. FIG 12 show3 the situation 
for ThC. Since the gamma lifetimes 
are on the order of 10 -5 times smaller 
than the alpha lifetimes for the ex- 
cited parent states, the long range 
alpha intensities are verv low. 

Half livesirf 

3 s 10-7 

r, a io-i4 

?=M 1Q-17 

Decay of ThC ' 
FIG III. 12 

F. Appendix to Gh.III 

The calculation of C + and C_ as used In III. 8 will be made 
here. See- page 62. The continuity conditions across the bound- 
ary at r = R give 

since ***.&& = *h(H) 




Ch. Ill 

At the virtual level E = E , C + = or 


which is less than unity .for an 
unstable nucleus 

Thus Y% -z;/n7f where n is an integer 



3E ft.* 

This may also be expressed In terras of J E by using III. 15, 




C + - If'// 


III. 16